Given the RGB components of a 4 ×4 color image, find the YCbCr components of its YCbCr model. *(

First, we need to convert the RGB components to the YCbCr components using the following formula

Given the RGB components of a 4 ×4 color image, find the...

Given the RGB components of a $4 \times 4$ color image, find the $\mathrm{YCbCr}$ components of its YCbCr model. *(i) $$ \left[\begin{array}{llll} 142 & 142 & 150 & 149 \\ 144 & 144 & 143 & 163 \\ 145 & 144 & 152 & 157 \\ 147 & 153 & 155 & 126 \end{array}\right]\left[\begin{array}{llll} 15 & 15 & 24 & 24 \\ 17 & 17 & 12 & 35 \\ 18 & 18 & 16 & 24 \\ 18 & 21 & 20 & 18 \end{array}\right]\left[\begin{array}{llll} 44 & 44 & 49 & 48 \\ 46 & 47 & 37 & 59 \\ 47 & 46 & 43 & 51 \\ 48 & 49 & 42 & 28 \end{array}\right] $$ (ii) $$ \left[\begin{array}{llll} 227 & 230 & 230 & 229 \\ 224 & 215 & 214 & 215 \\ 208 & 207 & 208 & 209 \\ 191 & 197 & 198 & 195 \end{array}\right]\left[\begin{array}{llll} 0 & 6 & 5 & 3 \\ 4 & 0 & 0 & 0 \\ 2 & 2 & 2 & 1 \\ 0 & 6 & 6 & 3 \end{array}\right]\left[\begin{array}{llll} 27 & 32 & 32 & 32 \\ 30 & 20 & 19 & 22 \\ 19 & 15 & 14 & 16 \\ 13 & 15 & 14 & 13 \end{array}\right] $$ (iii) $$ \left[\begin{array}{llll} 226 & 226 & 226 & 225 \\ 226 & 226 & 225 & 223 \\ 226 & 226 & 225 & 222 \\ 226 & 226 & 225 & 223 \end{array}\right]\left[\begin{array}{llll} 239 & 239 & 238 & 235 \\ 239 & 239 & 238 & 235 \\ 239 & 239 & 238 & 235 \\ 239 & 239 & 238 & 236 \end{array}\right]\left[\begin{array}{llll} 219 & 221 & 221 & 220 \\ 219 & 221 & 220 & 218 \\ 219 & 221 & 220 & 218 \\ 219 & 221 & 221 & 219 \end{array}\right] $$

Given the RGB components of a $4 \times 4$ color image, find the $\mathrm{YCbCr}$ components of its YCbCr model. *(i)
$$
\left[\begin{array}{llll}
142 & 142 & 150 & 149 \\
144 & 144 & 143 & 163 \\
145 & 144 & 152 & 157 \\
147 & 153 & 155 & 126
\end{array}\right]\left[\begin{array}{llll}
15 & 15 & 24 & 24 \\
17 & 17 & 12 & 35 \\
18 & 18 & 16 & 24 \\
18 & 21 & 20 & 18
\end{array}\right]\left[\begin{array}{llll}
44 & 44 & 49 & 48 \\
46 & 47 & 37 & 59 \\
47 & 46 & 43 & 51 \\
48 & 49 & 42 & 28
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
227 & 230 & 230 & 229 \\
224 & 215 & 214 & 215 \\
208 & 207 & 208 & 209 \\
191 & 197 & 198 & 195
\end{array}\right]\left[\begin{array}{llll}
0 & 6 & 5 & 3 \\
4 & 0 & 0 & 0 \\
2 & 2 & 2 & 1 \\
0 & 6 & 6 & 3
\end{array}\right]\left[\begin{array}{llll}
27 & 32 & 32 & 32 \\
30 & 20 & 19 & 22 \\
19 & 15 & 14 & 16 \\
13 & 15 & 14 & 13
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
226 & 226 & 226 & 225 \\
226 & 226 & 225 & 223 \\
226 & 226 & 225 & 222 \\
226 & 226 & 225 & 223
\end{array}\right]\left[\begin{array}{llll}
239 & 239 & 238 & 235 \\
239 & 239 & 238 & 235 \\
239 & 239 & 238 & 235 \\
239 & 239 & 238 & 236
\end{array}\right]\left[\begin{array}{llll}
219 & 221 & 221 & 220 \\
219 & 221 & 220 & 218 \\
219 & 221 & 220 & 218 \\
219 & 221 & 221 & 219
\end{array}\right]
$$

Key Concepts

Matrix Representation of Digital Images

Digital images are often expressed as matrices where each entry represents a pixel’s intensity value in a particular channel. Operations on these matrices, such as applying conversion formulas or other transformations, are key procedures in tasks like color space conversion, filtering, and other image processing techniques.

Luminance and Chrominance

In image processing, luminance (Y) provides the brightness or intensity information, while chrominance (Cb and Cr) carries the color difference data. This separation allows for more efficient compression and manipulation since the human visual system is less sensitive to changes in chrominance compared to luminance.

Color Space Conversion

Color space conversion is the process of transforming image data from one color model to another, such as from RGB to YCbCr. This involves applying a set of linear transformation equations to the RGB values, which recalculates the image’s brightness and color information into the Y, Cb, and Cr components.

RGB Color Model

The RGB color model represents images using three components—red, green, and blue. In this model, colors are formed by combining different intensities of these three primary colors. It is widely used in electronic displays, cameras, and image sensors, making it fundamental in digital image representation.

YCbCr Color Model

The YCbCr color model separates image information into a luminance component (Y) and two chrominance components (Cb and Cr) that represent color differences. This separation is advantageous for image and video compression since human vision is more sensitive to variations in brightness than in color, allowing for efficient data reduction.

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